Fitting a curve to a plurality of data points is commonly performed in engineering applications where characterization of the plurality of data points is required in order to obtain information regarding a response of a system. For example, curve fitting of data points is performed in the field of semiconductor processing such as in controlling focus for a photolithographic tool. Image size measurement data points generated from images produced by the photolithographic tool for various focus settings can be analyzed to fit a curve to the data points in order to determine a “best focus” condition for the photolithographic tool.
According to one conventional method, a 2nd-order polynomial (i.e. parabolic model) is fit to image size measurement data points. Unfortunately, parabolic modeling is resistant to classical point-tossing methodologies, usually based upon model residuals, making it difficult to detect and discard inaccurate measurement data. The reasons for this difficulty include:                a) parabolic models are weighted heavily towards their extremes because of the squaring of the independent variable. In other words, the outermost data points have the most influence on the modeled parameters; and        b) the outermost data points that have the most influence on the modeled parameters are also the most likely to be inaccurate. This is due to image quality being poor at the extremes of focus of the photolithographic tool, thus making the images difficult to measure.        
In order to improve confidence in the parabolic model, and to minimize the effects of inaccurate image size measurement data points, over-sampling of the photolithographic system occurs. In a manufacturing environment, this decreases productivity which increases manufacturing costs.
Another conventional method is to fit the image size measurement data to a 2nd-order polynomial and remove data points based upon the individual residual errors. Data at the extremes of focus of the photolithographic tool typically do not fit parabolic behavior due to poor image quality, thus making the images difficult to measure. Yet the 2nd-order polynomial method gives excessive weighting to the data at the extremes of focus resulting in a reduction in accuracy in the determination of best focus. For example, FIG. 1 illustrates a 2nd-order polynomial 30 fit to a plurality of measurement data points (denoted by X). Data points 35 at the extremes of focus are given excessive weighting such that data points 40 are erroneously removed and are not included in the determination of the 2nd-order polynomial 30.
Yet another conventional method is to fit the image size measurement data to a 4th-order polynomial or fit the image size measurement data to a 4th-order polynomial and remove data points based upon the individual residual errors. The 4th-order polynomial method accommodates the data points at the extremes of focus at the expense of the center data points where best focus is typically located. As described above with reference to the 2nd-order polynomial, data at the extremes of focus of the photolithographic tool typically do not fit parabolic behavior due to poor image quality. Thus, the 4th-order polynomial method also gives excessive weighting to the data at the extremes of focus resulting in a reduction in accuracy in the determination of best focus.
What is required is a method to fit a curve to a plurality of data points where the effect of a one or more inaccurate data points is reduced.